metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.1Dic7, C23⋊(C7⋊C8), C7⋊2(C23⋊C8), (C22×C14)⋊2C8, (C2×C28).227D4, (C23×C14).3C4, (C22×C28).1C4, (C22×C4).3D14, C28.55D4⋊21C2, (C22×C4).1Dic7, C14.16(C23⋊C4), C14.13(C22⋊C8), C14.6(C4.D4), C2.1(C28.D4), (C2×C14).22M4(2), C2.1(C23⋊Dic7), C23.20(C2×Dic7), C2.3(C28.55D4), C22.4(C4.Dic7), (C22×C28).324C22, C22.23(C23.D7), C22.2(C2×C7⋊C8), (C2×C14).29(C2×C8), (C2×C22⋊C4).1D7, (C2×C4).159(C7⋊D4), (C14×C22⋊C4).21C2, (C2×C14).85(C22⋊C4), (C22×C14).123(C2×C4), SmallGroup(448,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.Dic7
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=c, f2=ce7, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 308 in 98 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C23, C23, C14, C14, C22⋊C4, C2×C8, C22×C4, C24, C28, C2×C14, C2×C14, C22⋊C8, C2×C22⋊C4, C7⋊C8, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23⋊C8, C2×C7⋊C8, C7×C22⋊C4, C22×C28, C23×C14, C28.55D4, C14×C22⋊C4, C24.Dic7
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), Dic7, D14, C22⋊C8, C23⋊C4, C4.D4, C7⋊C8, C2×Dic7, C7⋊D4, C23⋊C8, C2×C7⋊C8, C4.Dic7, C23.D7, C28.55D4, C28.D4, C23⋊Dic7, C24.Dic7
►On 112 points
Generators in S112
(2 89)(4 91)(6 93)(8 95)(10 97)(12 99)(14 101)(16 103)(18 105)(20 107)(22 109)(24 111)(26 85)(28 87)(29 58)(30 44)(31 60)(32 46)(33 62)(34 48)(35 64)(36 50)(37 66)(38 52)(39 68)(40 54)(41 70)(42 56)(43 72)(45 74)(47 76)(49 78)(51 80)(53 82)(55 84)(57 71)(59 73)(61 75)(63 77)(65 79)(67 81)(69 83)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 57)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 85)(27 86)(28 87)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 53 22 46 15 39 8 32)(2 38 23 31 16 52 9 45)(3 51 24 44 17 37 10 30)(4 36 25 29 18 50 11 43)(5 49 26 42 19 35 12 56)(6 34 27 55 20 48 13 41)(7 47 28 40 21 33 14 54)(57 106 78 99 71 92 64 85)(58 91 79 112 72 105 65 98)(59 104 80 97 73 90 66 111)(60 89 81 110 74 103 67 96)(61 102 82 95 75 88 68 109)(62 87 83 108 76 101 69 94)(63 100 84 93 77 86 70 107)
G:=sub<Sym(112)| (2,89)(4,91)(6,93)(8,95)(10,97)(12,99)(14,101)(16,103)(18,105)(20,107)(22,109)(24,111)(26,85)(28,87)(29,58)(30,44)(31,60)(32,46)(33,62)(34,48)(35,64)(36,50)(37,66)(38,52)(39,68)(40,54)(41,70)(42,56)(43,72)(45,74)(47,76)(49,78)(51,80)(53,82)(55,84)(57,71)(59,73)(61,75)(63,77)(65,79)(67,81)(69,83), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,57)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,85)(27,86)(28,87)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,53,22,46,15,39,8,32)(2,38,23,31,16,52,9,45)(3,51,24,44,17,37,10,30)(4,36,25,29,18,50,11,43)(5,49,26,42,19,35,12,56)(6,34,27,55,20,48,13,41)(7,47,28,40,21,33,14,54)(57,106,78,99,71,92,64,85)(58,91,79,112,72,105,65,98)(59,104,80,97,73,90,66,111)(60,89,81,110,74,103,67,96)(61,102,82,95,75,88,68,109)(62,87,83,108,76,101,69,94)(63,100,84,93,77,86,70,107)>;
G:=Group( (2,89)(4,91)(6,93)(8,95)(10,97)(12,99)(14,101)(16,103)(18,105)(20,107)(22,109)(24,111)(26,85)(28,87)(29,58)(30,44)(31,60)(32,46)(33,62)(34,48)(35,64)(36,50)(37,66)(38,52)(39,68)(40,54)(41,70)(42,56)(43,72)(45,74)(47,76)(49,78)(51,80)(53,82)(55,84)(57,71)(59,73)(61,75)(63,77)(65,79)(67,81)(69,83), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,57)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,85)(27,86)(28,87)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,53,22,46,15,39,8,32)(2,38,23,31,16,52,9,45)(3,51,24,44,17,37,10,30)(4,36,25,29,18,50,11,43)(5,49,26,42,19,35,12,56)(6,34,27,55,20,48,13,41)(7,47,28,40,21,33,14,54)(57,106,78,99,71,92,64,85)(58,91,79,112,72,105,65,98)(59,104,80,97,73,90,66,111)(60,89,81,110,74,103,67,96)(61,102,82,95,75,88,68,109)(62,87,83,108,76,101,69,94)(63,100,84,93,77,86,70,107) );
G=PermutationGroup([[(2,89),(4,91),(6,93),(8,95),(10,97),(12,99),(14,101),(16,103),(18,105),(20,107),(22,109),(24,111),(26,85),(28,87),(29,58),(30,44),(31,60),(32,46),(33,62),(34,48),(35,64),(36,50),(37,66),(38,52),(39,68),(40,54),(41,70),(42,56),(43,72),(45,74),(47,76),(49,78),(51,80),(53,82),(55,84),(57,71),(59,73),(61,75),(63,77),(65,79),(67,81),(69,83)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,57),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,112),(26,85),(27,86),(28,87),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,53,22,46,15,39,8,32),(2,38,23,31,16,52,9,45),(3,51,24,44,17,37,10,30),(4,36,25,29,18,50,11,43),(5,49,26,42,19,35,12,56),(6,34,27,55,20,48,13,41),(7,47,28,40,21,33,14,54),(57,106,78,99,71,92,64,85),(58,91,79,112,72,105,65,98),(59,104,80,97,73,90,66,111),(60,89,81,110,74,103,67,96),(61,102,82,95,75,88,68,109),(62,87,83,108,76,101,69,94),(63,100,84,93,77,86,70,107)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | Dic7 | D14 | Dic7 | C7⋊D4 | C7⋊C8 | C4.Dic7 | C23⋊C4 | C4.D4 | C28.D4 | C23⋊Dic7 |
| kernel | C24.Dic7 | C28.55D4 | C14×C22⋊C4 | C22×C28 | C23×C14 | C22×C14 | C2×C28 | C2×C22⋊C4 | C2×C14 | C22×C4 | C22×C4 | C24 | C2×C4 | C23 | C22 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C24.Dic7 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 49 |
| 0 | 0 | 0 | 0 | 49 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[32,0,0,0,0,0,0,60,0,0,0,0,0,0,0,30,0,0,0,0,30,0,0,0,0,0,0,0,0,49,0,0,0,0,49,0],[0,112,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C24.Dic7 in GAP, Magma, Sage, TeX
C_2^4.{\rm Dic}_7 % in TeX
G:=Group("C2^4.Dic7"); // GroupNames label
G:=SmallGroup(448,82);
// by ID
G=gap.SmallGroup(448,82);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,100,1123,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=c,f^2=c*e^7,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^13>;
// generators/relations